High entropy oxides are attracting attention for catalysis, but there are relatively few detailed studies on their precise structure, hampering true detailed studies on fundamental properties affecting their activities. In addition, diffusion has been often characterized as generally slow in high-entropy systems. Here, we determine the precise oxygen content and structure of the fluorite-like high-entropy oxide (La,Ce,Pr,Nd,Y)O_1.68, and have identified a large oxygen storage capacity based on efficient Ce/Pr redox due to facile oxide diffusion pathways and suppression of sintering. The structure and composition were identified through a combined Rietveld refinement of X-ray and neutron diffraction data, and the oxidation state of Ce and Pr were investigated by HERFD-XANES. (La,Ce,Pr,Nd,Y)O_1.68 utilizes the full redox range of Ce/Pr, resulting in a high oxygen storage cumulative capacity despite the lower content of Ce/Pr compared to other well-known ceria derivatives. Diffusion pathway analysis by bond valence site energy mapping shows decreased barriers for oxide anion diffusion through the bulk, also benefiting redox reactions. The high-entropy nature also suppresses sintering, resulting in better cycling performance. This results in a higher performance as a methane oxidation catalyst support. We also investigate its use as a NO_x reduction catalyst support.